Linear-Response TDDFT in Frequency-Reciprocal space on a...
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Linear-Response TDDFTin Frequency-Reciprocal space
on a Plane-Waves basis:the DP (Dielectric Properties) code
Valerio Olevano,
Lucia Reining, Francesco Sottile and Silvana Botti
September 1, 2004
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Why and Wherethe Frequency-Reciprocal space
could be convenient
• Reciprocal Space⇒ Infinite Periodic Systems (Bulk, but also Surfaces,
Wires, Tubes with the use of Supercells)
• Frequency Space ⇒ Spectra
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Outlook
• Motivation
• TDDFT
• Linear Response TDDFT
• Frequency-Reciprocal space TDDFT
• TDDFT on a PW basis: the DP code
• Approximations and Results
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Optical Absorption
2.5 3 3.5 4 4.5 5 5.5
ω [eV]
0
10
20
30
40
50
Im ε
(ω)
EXP
SiliconOptical Absorption
Lautenschlager etal., PRB 36, 4821(1987).
Aspnes and Studna,PRB 27, 985 (1983).
Refraction Indices
1.0 2.0 3.0 4.0 5.0 6.0 7.0
ω [eV]
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
n, k
n EXPk EXP
SiliconRefraction Indices
Reflectance
1 2 3 4 5 6 7
ω [eV]
0.3
0.4
0.5
0.6
0.7
0.8
R
EXP
SiliconReflectance
Aspnes and Studna,PRB 27, 985 (1983).
Stiebling, Z. Phys. B31, 355 (1978).
Energy-Loss (EELS)
10 15 20 25
energy loss, ω [eV]
0.0
1.0
2.0
3.0
4.0
5.0
- Im
ε-1
(ω)
EXP
SiliconEELS
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Inelastic X-ray Scattering Spectroscopy (IXXS)
0.0 20.0 40.0 60.0
ω [eV]
0.0
0.2
0.4
-Im ε
-1(q
,ω)
EXP
SiliconIXSS Dynamic Structure Factor, q = 1.25 a.u. || [111]
Sturm et al., PRL 48,1425 (1982).
Schulke andKaprolat, PRL 67,879 (1991).
Coherent Inelastic Scattering Spectroscopy (CIXS)
0.0 10.0 20.0 30.0 40.0
ω [eV]
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
- Im
ε-1
(q,q
+G,ω
)
EXP
SiliconCIXS, q = (0.83,0.33,0.33) 2π/a, G = (-1,-1,-1) 2π/a
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Problems for the Theory
• Reproduce Experimental Spectra
• Offer Reference Spectra to the Experiment
• Predict Optical and Dielectric Properties ⇒ Theoretical Spectroscopy
Facility
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Accessed Observable:the Macroscopic Dielectric Function εM(q, ω)
ε∞ = εM(q = 0, ω = 0) dielectric constant
ABS = Im εM(q = 0, ω) optical absorption
EEL = −Im1
εM(q, ω)energy loss
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What is the TDDFT?
• TDDFT is an extension of DFT; it is a DFT with time-dependent
external potential:
v(r) → v(r, t)
• Milestones of TDDFT:
– Runge, Gross (1984): rigorous basis of TDDFT.
– Gross, Kohn (1985): TDDFT in Linear Response.
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DFT vs TDDFT
Hohenberg-Kohn:
v(r) ⇔ ρ(r)Runge-Gross:
v(r, t) ⇔ ρ(r, t)
The Total Energy:
〈Φ|H|Φ〉 = E[ρ]The Action:∫ t1
t0dt 〈Φ(t)|i ∂
∂t − H(t)|Φ(t)〉 = A[ρ]
are unique functionals of the density.
The stationary points of:
the Total EnergyδE[ρ]δρ(r) = 0
the Action:δA[ρ]δρ(r,t) = 0
give the exact density of the system:
ρ(r) ρ(r, t)
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DFT vs TDDFT
Kohn-Sham:ρ(r)=
PNi=1|φKS
i (r)|2
vKS(r)=v(r)+R
dr′ ρ(r′)|r−r′|+
δExc[ρ]δρ(r)
HKS(r)φKSi (r)=εKS
i φKSi (r)
Runge-Gross:ρ(r,t)=
PNi=1|φKS
i (r,t)|2
vKS(r,t)=v(r,t)+R
dr′ ρ(r′,t)|r−r′|+
δAxc[ρ]δρ(r,t)
i ∂∂tφ
KSi (r,t)=HKS(r,t)φKS
i (r,t)
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TDDFT in Linear ResponseGross and Kohn (1985)
If:
vext(r, t) = vext(r) + δvext(r, t)
with:
δvext(r, t) � vext(r)
then:
TDDFT = DFT + Linear Response(to the time-dependent perturbation δvext)
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Hohenberg-Kohn Theoremfor Linear Response TDDFT
DFT: vext(r) ⇔ ρ(r)
TDDFT: vext(r, t) ⇔ ρ(r, t) Runge-Gross theorem
LR-TDDFT: vext(r) + δvext(r, t) ⇔ ρ(r) + δρ(r, t)⇓
δvext(r, t) ⇔ δρ(r, t)
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Linear Response TDDFTcalculation scheme
1. Ordinary DFT calculation:
vext(r) ⇒ ρ(r), εKS, φKS(r)
2. Linear Response calculation:
δvext(r, t) ⇒ δρ(r, t)
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Polarizability χ
δvext external perturbation
δρ induced density
Definition of the polarizability χ:
δρ = χδvext
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Variation in the Total Potential
The variation in the density induces a variation in the Hartree and in the exchange-
correlation potentials which screen the external perturbation:
δvH =δvH
δρδρ = vcδρ
δvxc =δvxc
δρδρ = fxcδρ
so that the variation in the total potential (external + screening) is
δvtot = δvext + δvH + δvxc
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Exchange-Correlation Kernel fxc
The exchange-correlation kernel is defined:
fxcdef=
δvxc
δρ
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LR-TDDFT Kohn-Sham scheme:the Independent Particle Polarizability χ(0)
Let’s introduce a ficticious s, Kohn-Sham non-interacting system such that:
δρs = δρ
Then, instead of calculating χ, we can more easily calculate the polarizability χ(0) (also
χs or χKS) of this non-interacting system, called independent particle polarizability
and defined:
δρ = χ(0)δvtot
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Independent Particle Polarizability χ(0)
By variation δvtot (= δvs = δvKS) of the Kohn-Sham equations, one obtains theLinear Response variation of the density δρ and then an expression for χ(0) in termsof the Kohn-Sham energies and wavefunctions:
χ(0)(r, r′, ω) = 2∑i 6=j
(fi − fj)φi(r)φ∗j(r)φ
∗i (r
′)φj(r′)εi − εj − ω − iη
(Adler and Wiser)
In Frequency Reciprocal space:
χ(0)GG′(q, ω) = 2
∑i 6=j
(fi − fj)〈φj|e−i(q+G)r|φi〉〈φi|ei(q+G′)r|φj〉
εi − εj − ω − iη
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χ as a function of χ(0)
From:{δρ = χδvext
δρ = χ(0)δvtot
the polarizability in term of the independent particle polarizability is:
χ = (1− χ(0)vc − χ(0)fxc)−1χ(0)
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Dielectric Function ε
Definition of the dielectric function:
δvtot = ε−1δvext
ε−1 = 1 + vcχ
In a periodic system it has this form:
εGG′(q, ω)
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Macroscopic Dielectric Function
Definition:
εM(q, ω) def=1
ε−100 (q, ω)
Approximation: Neglecting Local Fields:
εNLFM (q, ω) = ε00(q, ω)
Macroscopic dielectric constant:
ε∞ = limq→0
εM(q, ω = 0)
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Local Fields
Effect of non-diagonal elements (inhomogeneities):
δvtotG =
∑G′
ε−1GG′ δv
extG′
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LR-TDDFT Calculation SchemeResume
ρDFT
"""b"b
"b"b
"b
φDFTKS
##GGGGGGGG
|| |<|<
|<|<
|<
εDFTKS
||xxxxxxxx
fxc
++
χ(0)
��
χ
��
ε
��
εM
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the code
• Definition: Linear-Response TDDFT code in Frequency-Reciprocal space on a PW basis
• Purposes: DP calculates Dielectric and Optical Properties (Absorption, Reflectivity, Refraction
indices, EELS, IXSS, CIXS, etc.) for Bulk systems, Surfaces, Cluster, Moleculs, Atoms (through
Supercells) made of Insulator, Semiconductor and Metallic elements.
• Approximations: RPA, ALDA, GW-RPA, LRC, non-local kernels, Mapping Theory, etc., with
and without Local Fields (LF)
• Languages: Fortran90 with C insertions (shell, parser, some libraries); Vectorialized and Partially
Parallelized (MPI)
• Machines: PC-Linux IFC, Compaq/HP True64, IBM AIX, SG IRIX, NEC SX5, Fujitsu
• Libraries: BLAS, Lapack, FFTW, CXML, ESSL, ASL, Nag, Goedecker, MFFT
• Interfaces: LSI-CP, ABINIT, PWSCF, FHIMD
DP package, Copyright 1998-2004 Valerio Olevano, Lucia Reining, Francesco Sottile, CNRS.
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DP Flow Diagram
εKSi
��
φKSi (G)
FFT−1��
interface
φKSi (r)
��
ρij(r) = φ∗i (r)φj(r)
FFT��
ρij(G)
��
χ(0)GG′(q, ω) =
∑ij(fi − fj)
ρij(G)ρ∗ij(G′)
εi−εj−ω
INV��
fxc// χGG′(q, ω)
��
ε−1GG′(q, ω)
sshhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
++VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV
εLFM (q, ω) εNLF
M (q, ω)
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DP Tricks
If we only need ε−100 = 1− v0χ00, that is only χ00
then, instead of solving (inverting a full matrix):
χGG′ =(1− χ(0)vc − χ(0)fxc
)−1
GG′′χ
(0)G′′G′
we solve the linear system for only the first column of χG′0:(1− χ(0)vc − χ(0)fxc
)GG′
χG′0 = χ(0)G0
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DP Performances:CPU scaling and Memory usage
• CPU scaling for χ(0): N2G ·Nk ·Nb ·Nr log Nr
• CPU scaling for ε−1: N2G ·Nω
• Memory occupation: N2G ·Nω + Nr ·Nk ·Nb [sizeofcomplex]
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Exchange-Correlation Kernel fxc
and its Approximations
Definition of the exchange-correlation kernel:
fxcdef=
δvxc
δρ
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RPA Approximation
Random Phase Approximation (RPA) = Neglect of the exchange-correlation
effects:
RPA: fxc = 0
εRPA = 1− vcχ(0)
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ALDA Approximation (TDLDA)
Adiabatic Local Density Approximation:
ALDA: fALDAxc =
δvLDAxc
δρ
∣∣∣∣ω=0
fALDAxc (r, r′) = A(r)δ(r, r′) local in r-space
fALDAxc GG′(q) = B(G−G′)
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Results on EELS
10 20 30 40energy [eV]
0
2
loss
fun
ctio
n
RPA LFRPA NLFEXPTDLDA
GraphiteEELS
Θ = 90ο
Θ = 45ο
Θ = 30ο
Θ = 0ο
Θ = 77ο
in plane
out of plane
A. Marinopoulos et al., Phys. Rev. B 69, 245419 (2004).
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Results on IXSS and CIXS
0 10 20 30 40 50 60
ω [eV]
0.0
0.2
0.4
0.6
-Im ε
-1(q
,ω)
EXPRPATDLDA
SiliconIXSS Dynamic Structure Factor, q = 1.25 a.u. || [111]
0.0 10.0 20.0 30.0 40.0
ω [eV]
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
- Im
ε-1
(q,q
+G,ω
)RPATDLDAEXP
SiliconCIXS, q = (0.83,0.33,0.33) 2π/a, G = (-1,-1,-1) 2π/a
V. Olevano, PhD thesis (1999).
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Conclusions on EELS
• RPA is good but with LF (Local Fields)
• ALDA does not improve unambigously
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Results on ABS
3 4 5 6
ω [eV]
0
10
20
30
40
50
60
ε 2(ω)
EXPRPATDLDARPA NLF
SiliconOptical Absorption
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LRC Approximation
Long-Range Contribution only:
fLRCxc = − α
(q + G)2
α = 4.6ε−1∞ − 0.2
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Results on ABS
3 4 5 6
ω [eV]
0
10
20
30
40
50
60
ε 2(ω)
EXPRPATDLDATDDFT LRC
SiliconOptical Absorption
L. Reining et al., Phys. Rev. Lett. 88, 066404 (2002).
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Mapping Theory (MT)
Mapping BSE on TDDFT:
fMPxc [{φi}, {εi}]
χ = χ(0)(χ(0) − χ(0)vcχ
(0) − T [{φi}, {εi}])−1
χ(0)
T = χ(0)fxcχ(0)
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Results on Solid Argon ABS
11 12 13 14 150
5
10
15Exp.BSETDLDAT2
F. Sottile et al., to be published
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Conclusions on Optical Properties
• RPA and ALDA fail
• LRC already reproduces small Excitonic Effects
• Mapping Theory should be used for Strong Excitons
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DP licence
• DP costed 7 years human full-time to the authors: is GNU/GPL going to reducethe charge on the central?
• developping software is a job in itself!
• who is going to pay for this? are private companies interested in this software? notyet!
• if the scientific community is interested and recognize the importance, then it musthire people to develop these codes!
• International, European or even National institutions must be setup to managescientific codes interesting for the whole international scientific community.
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DP licence and ETSF
IF ( )
THEN
= OpenSource GNU/GPL